The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on a possible form of constraints may affect the complexity, and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming to separate ... more >>>

A wide variety of combinatorial problems can be represented in the form of the Constraint Satisfaction Problem (CSP). The general CSR is known to be NP-complete, however, some restrictions on the possible form of constraints may lead to a tractable subclass. Jeavons and coauthors have shown that the complexity of ... more >>>

We prove a strong inapproximability result for routing on directed
graphs with low congestion. Given as input a directed graph on $N$
vertices and a set of source-destination pairs that can be connected
via edge-disjoint paths, we prove that it is hard, assuming NP
doesn't have $n^{O(\log\log n)}$ time randomized ... more >>>

The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite
relational structure H can be expressed as follows: given a
relational structure G over the same vocabulary,
determine the number of homomorphisms from G to H.
In this paper we characterize relational structures H for which
#CSP(H) can be solved in ... more >>>

An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least $(1-g(\varepsilon))$-fraction of the constraints given a $(1-\varepsilon)$-satisfiable instance, where $g(\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$, $g(0)=0$.
Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction ... more >>>

We study two natural extensions of Constraint Satisfaction Problems (CSPs). {\em Balance}-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of {\em Hard}-Max-CSP consists of {\em soft constraints} and {\em hard constraints}, and the goal is to maximize ... more >>>

A graph $G$ has an \emph{$S$-factor} if there exists a spanning subgraph $F$ of $G$ such that for all $v \in V: \deg_F(v) \in S$.
The simplest example of such factor is a $1$-factor, which corresponds to a perfect matching in a graph. In this paper we study the computational ... more >>>

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on $n$ variables taking values in $\{0,\ldots,q-1\}$, we prove that improving over the trivial approximability by a factor of $q$ requires $\Omega(n)$ space even on instances with $O(n)$ constraints. We also identify ... more >>>

A Boolean maximum constraint satisfaction problem, Max-CSP\((f)\), is specified by a predicate \(f:\{-1,1\}^k\to\{0,1\}\). An \(n\)-variable instance of Max-CSP\((f)\) consists of a list of constraints, each of which applies \(f\) to \(k\) distinct literals drawn from the \(n\) variables. For \(k=2\), Chou, Golovnev, and Velusamy [CGV20, FOCS 2020] obtained explicit ratios ... more >>>

The study of the complexity of the constraint satisfaction problem (CSP), centred around the Feder-Vardi Dichotomy Conjecture, has been very prominent in the last two decades. After a long concerted effort and many partial results, the Dichotomy Conjecture has been proved in 2017 independently by Bulatov and Zhuk.

At about ... more >>>

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits:

- For every $k \geq 1$, there is a monotone function in ${\rm AC^0}$ (constant-depth poly-size circuits) that requires monotone circuits of depth $\Omega(\log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai ... more >>>

The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only $(1-\varepsilon)$-satisfiable (where the parameter is the number of variables) for some constant $0<\varepsilon<1$.

We ... more >>>

The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an $\varepsilon$ fraction of constraints for some absolute constant $\varepsilon > 0$. PIH plays the role of ... more >>>